Ultra-luminous infrared galaxies (ULIRGs) have infrared luminosities $L_{\mathrm{IR}} ≥ 10^{12} L_{⊙}$, making them the most luminous objects in the infrared sky. These dusty objects are generally powered by starbursts with star-formation rates that exceed $100~ M_{⊙}~ \mathrm{yr}^{-1}$, possibly combined with a contribution from an active galactic nucleus. Such environments make ULIRGs plausible sources of astrophysical high-energy neutrinos, which can be observed by the IceCube Neutrino Observatory at the South Pole. We present a stacking search for high-energy neutrinos from a representative sample of 75 ULIRGs with redshift $z ≤ 0.13$ using 7.5 years of IceCube data. The results are consistent with a background-only observation, yielding upper limits on the neutrino flux from these 75 ULIRGs. For an unbroken $E^{-2.5}$ power-law spectrum, we report an upper limit on the stacked flux $Φ_{ν_μ+ \barν_μ}^{90\%} = 3.24 \times 10^{-14}~ \mathrm{TeV^{-1}~ cm^{-2}~ s^{-1}}~ (E/10~ \mathrm{TeV})^{-2.5}$ at 90% confidence level. In addition, we constrain the contribution of the ULIRG source population to the observed diffuse astrophysical neutrino flux as well as model predictions.